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I am trying to find the limit of:

$\lim_{\sigma^2 \to 0} \int_{\Omega} \frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{||\tilde{y} - y ||^2}{2 \sigma^2}} \frac{1}{\sigma^2} (\tilde{y} - y) dy$

i.e. the integral of the derivative of the Normal as $\sigma^2$ goes to zero.

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It isn't homework, is it? – Harald Hanche-Olsen Nov 13 2009 at 1:21
Actually I would also be interested in how the construction of the Dirac delta distribution as the limit of the Gaussian works. – suppe Nov 13 2009 at 1:24
this looks a bit like homework doesn't it? – ioannis.parissis Nov 13 2009 at 1:27
If it is a standard homework problem could you point me to a book with the tools required to attack this problem. This construct pops up in the first variation of roughly something like $\int_{\Omega} || E( Y|f(y) = x ) - y ||^2 dy $. I expressed the conditional expectation with Dirac delta and since I did not know how to handle it in the first variation and tried to replace with the Gaussian now i face the problem to come up with a meaning for my first variation and this object pops up. Any pointers are appreciated. – suppe Nov 13 2009 at 1:35
Also I know how to get to the distributional derivative of the dirac delta via integration by parts. But in my case I have something like $\int_{\Omega} \delta(f(\tilde(y) - f(y)) p(y) y dy$ for which I can't make sense with integration by parts if I take the first variation of $f$. – suppe Nov 13 2009 at 2:00
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closed as too localized by Qiaochu Yuan, David Speyer, Scott Morrison♦ Nov 13 2009 at 2:21

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